ESBO-ETC/esbo_etc/classes/psf/Airy.py

271 lines
14 KiB
Python

from typing import Union
import numpy as np
from astropy import units as u
from .IPSF import IPSF
from ..sensor.PixelMask import PixelMask
from scipy.optimize import newton, fmin, bisect
from scipy.special import j0, j1
from scipy.signal import fftconvolve
from scipy.integrate import quad
from scipy.interpolate import interp1d
class Airy(IPSF):
"""
A class for modelling the PSF using an airy disk.
"""
@u.quantity_input(wl="length", d_aperture="length")
def __init__(self, f_number: float, wl: u.Quantity, d_aperture: u.Quantity, osf: float, pixel_size: u.Quantity):
"""
Initialize a new PSF from a airy disk.
Parameters
----------
f_number : float
The working focal number of the optical system
wl : Quantity
The central wavelength which is used for calculating the PSF
d_aperture : Quantity
The diameter of the telescope's aperture.
osf : float
The oversampling factor to be used for oversampling the PSF with regards to the pixel size.
pixel_size : Quantity
The size of a pixel as length-quantity.
"""
self.__f_number = f_number
self.__wl = wl
self.__d_aperture = d_aperture
self.__osf = osf
self.__pixel_size = pixel_size
self.__psf_jitter = None
def calcReducedObservationAngle(self, contained_energy: Union[str, int, float],
jitter_sigma: u.Quantity = None, obstruction: float = 0.0) -> u.Quantity:
"""
Calculate the reduced observation angle in lambda / d_ap for the given contained energy.
Parameters
----------
contained_energy : Union[str, int, float]
The percentage of energy to be contained within a circle with the diameter reduced observation angle.
jitter_sigma : Quantity
Sigma of the telescope's jitter in arcsec
obstruction : float
The central obstruction as ratio A_ob / A_ap
Returns
-------
reduced_observation_angle: Quantity
The reduced observation angle in lambda / d_ap
"""
# Calculate the reduced observation angle in lambda / D for the given encircled energy
reduced_observation_angle = 0 * u.dimensionless_unscaled
if type(contained_energy) == str:
# Encircled energy is of type string
if contained_energy.lower() == "peak":
# For the peak value of the PSF, the observation angle becomes zero which leads to one exposed
# pixel later in the code
reduced_observation_angle = 0 * u.dimensionless_unscaled
elif contained_energy.lower() == "fwhm":
# Width of the FWHM of the airy disk
reduced_observation_angle = 1.028
if not np.isclose(obstruction, 0.0):
# Use obstructed airy disk
reduced_observation_angle = newton(lambda y: self.__airy(np.pi * y, np.sqrt(obstruction)) - 0.5,
reduced_observation_angle / 2) * 2
contained_energy = "fwhm"
elif contained_energy.lower() == "min":
# Width of the first minimum of the airy disk
reduced_observation_angle = 1.22 * 2
contained_energy = 0.8377 * u.dimensionless_unscaled
if not np.isclose(obstruction, 0.0):
# Use obstructed airy disk
reduced_observation_angle = fmin(lambda y: self.__airy(np.pi * y, np.sqrt(obstruction)),
reduced_observation_angle / 2, disp=False)[0] * 2
contained_energy = self.__airy_int(np.pi * reduced_observation_angle / 2,
np.sqrt(obstruction)) * u.dimensionless_unscaled
else:
# Calculate the width numerically from the integral of the airy disk
contained_energy = contained_energy / 100 * u.dimensionless_unscaled
reduced_observation_angle = 2 * bisect(
lambda y: self.__airy_int(np.pi * y, np.sqrt(obstruction)) - contained_energy.value, 0, 100)
if jitter_sigma is not None and (isinstance(contained_energy, u.Quantity) or isinstance(contained_energy, str)
and contained_energy.lower() == "fwhm"):
# Convert jitter to reduced observation angle in lambda / d_ap
jitter_sigma = jitter_sigma.to(u.rad).value * self.__d_aperture / self.__wl.to(u.m)
# Calculate necessary grid length to accommodate the psf and 3-sigma of the gaussian
grid_width = (reduced_observation_angle / 2 + 3 * jitter_sigma.value)
# Calculate the reduced observation angle of a single detector pixel
reduced_observation_angle_pixel = (self.__pixel_size / (
self.__f_number * self.__d_aperture) * self.__d_aperture / self.__wl).decompose()
# Calculate the width of each grid element
dx = reduced_observation_angle_pixel.value / self.__osf
# Calculate the necessary number of points on the grid
n_points = np.ceil(grid_width / dx)
# Calculate the corresponding x-coordinates of each grid element
x = np.arange(1, n_points + 1) * dx
# Calculate the psf from an airy disk for each element on the grid
psf = self.__airy(np.pi * x, np.sqrt(obstruction))
# Calculate the integral of the undisturbed airy disk in order to scale the result of the convolution
# This corresponds to an integration in polar coordinates: I(r)*r dr dphi
total = np.sum(psf * x) * dx * 2 * np.pi
# Mirror the PSF to the negative x-domain
psf = np.concatenate((np.flip(psf), np.array([1]), psf))
# Calculate a gaussian kernel
kernel = 1 / (2 * np.pi * jitter_sigma.value ** 2) * np.exp(- x ** 2 / (2 * jitter_sigma.value ** 2))
# Mirror the kernel to the negative x-domain
kernel = np.concatenate((np.flip(kernel), np.array([1 / (2 * np.pi * jitter_sigma.value ** 2)]), kernel))
# Normalize the kernel
kernel = kernel / np.sum(kernel)
# Convolve the PSF with gaussian kernel
psf = fftconvolve(psf, kernel, mode="full")
# Reduce the PSF to the positive x-domain
psf = psf[int((psf.shape[0] - 1) / 2):]
# Scale the integral of the disturbed PSF equal to the undisturbed PSF
# This corresponds to an integration in polar coordinates: I(r)*r dr dphi
psf = psf / (np.sum(psf * np.arange(psf.shape[0]) * dx) * dx * 2 * np.pi) * total
self.__psf_jitter = psf
if isinstance(contained_energy, str) and contained_energy.lower() == "fwhm":
reduced_observation_angle = np.argmax(psf < psf[0] / 2) * reduced_observation_angle_pixel.value / \
self.__osf * 2
else:
# Calculate the rolling integral of the PSF
# This corresponds to an integration in polar coordinates: I(r)*r dr dphi
psf_int = np.cumsum(psf * np.arange(psf.shape[0]) * dx) * dx * 2 * np.pi
# Calculate the percentage of encircled energy by dividing by the infinite integral of the undisturbed
# airy disk
psf_int = psf_int / (4 / np.pi * 1 / (1 - obstruction))
# Calculate the reduced observation angle
reduced_observation_angle = np.argmax(psf_int > contained_energy) * dx * 2
return reduced_observation_angle * u.dimensionless_unscaled
@staticmethod
def __airy(x: Union[float, np.ndarray], obstruction: float = None) -> Union[float, np.ndarray]:
"""
Calculate function values of the airy disk
Parameters
----------
x : Union[float, np.ndarray]
radial coordinate to calculate the function value for.
obstruction : float
The linear central obstruction ratio of the aperture.
Returns
-------
res : Union[float, np.ndarray]
The function values of the airy disk at the given coordinates
"""
# Standardize input values
if not isinstance(x, np.ndarray):
x = np.array([x])
# Initialize return values and assign values for the singularity at x=0
res = np.zeros(x.shape)
res[np.isclose(x, 0.0)] = 1.0
x_temp = x[np.invert(np.isclose(x, 0.0))]
if obstruction and not np.isclose(obstruction, 0.0):
# Use obstructed airy disk
# See also https://en.wikipedia.org/wiki/Airy_disk#Obscured_Airy_pattern
res[np.invert(np.isclose(x, 0.0))] = 1 / (1 - obstruction ** 2) ** 2 * (
2 * (j1(x_temp) - obstruction * j1(obstruction * x_temp)) / x_temp) ** 2
else:
# Use unobstructed airy disk
# See also https://en.wikipedia.org/wiki/Airy_disk#Mathematical_formulation
res[np.invert(np.isclose(x, 0.0))] = (2 * j1(x_temp) / x_temp) ** 2
# Unbox arrays of length 1
if len(res.shape) == 1 and len(res) == 1:
res = res[0]
return res
@staticmethod
def __airy_int(x: float, obstruction: float = None) -> float:
"""
Calculate the integral of the airy disk from 0 to x.
Parameters
----------
x : float
The upper limit for the integration.
obstruction : float
The linear central obstruction ratio of the aperture.
Returns
-------
res : float
The integral of the airy disk.
"""
if np.isclose(x, 0.0):
# Short circuit for an integration-range of length 0
return 0.
else:
if obstruction and not np.isclose(obstruction, 0.0):
# Use integral of obstructed airy disk
# See also https://en.wikipedia.org/wiki/Airy_disk#Obscured_Airy_pattern
return 1 / (1 - obstruction ** 2) * (1 - j0(x) ** 2 - j1(x) ** 2 + obstruction ** 2 * (
1 - j0(obstruction * x) ** 2 - j1(obstruction * x) ** 2) - 4 * obstruction * quad(
lambda y: j1(y) * j1(obstruction * y) / y, 0, x, limit=100, epsrel=1e-6)[0])
else:
# Use unobstructed airy disk
# See also https://en.wikipedia.org/wiki/Airy_disk#Mathematical_formulation
return 1 - j0(x) ** 2 - j1(x) ** 2
def mapToPixelMask(self, mask: PixelMask, jitter_sigma: u.Quantity = None, obstruction: float = 0.0) -> PixelMask:
"""
Map the integrated PSF values to a sensor grid.
Parameters
----------
obstruction
mask : PixelMask
The pixel mask to map the values to. The values will only be mapped onto entries with the value 1.
jitter_sigma : Quantity
Sigma of the telescope's jitter in arcsec
obstruction : float
The central obstruction as ratio A_ob / A_ap
Returns
-------
mask : PixelMask
The pixel mask with the integrated PSF values mapped onto each pixel.
"""
# Calculate the indices of all non-zero elements of the mask
y_ind, x_ind = np.nonzero(mask)
# Extract a rectangle containing all non-zero values of the mask
mask_red = mask[y_ind.min():(y_ind.max() + 1), x_ind.min():(x_ind.max() + 1)]
# Calculate the new PSF-center indices of the reduced mask
psf_center_ind = [mask.psf_center_ind[0] - y_ind.min(), mask.psf_center_ind[1] - x_ind.min()]
# Oversample the reduced mask
mask_red_os = self._rebin(mask_red, self.__osf).view(PixelMask)
# Calculate the new PSF-center indices of the reduced mask
psf_center_ind = [(x + 0.5) * self.__osf - 0.5 for x in psf_center_ind]
reduced_observation_angle_pixel = (mask.pixel_size / (
self.__f_number * self.__d_aperture) * self.__d_aperture / self.__wl).decompose()
x_mesh, y_mesh = np.meshgrid(
(np.arange(mask_red_os.shape[1]) - psf_center_ind[
1]) * reduced_observation_angle_pixel.value / self.__osf,
(np.arange(mask_red_os.shape[0]) - psf_center_ind[
0]) * reduced_observation_angle_pixel.value / self.__osf)
dist = np.sqrt(x_mesh ** 2 + y_mesh ** 2)
if jitter_sigma is None:
res = self.__airy(dist * np.pi, np.sqrt(obstruction))
else:
if self.__psf_jitter is None:
self.calcReducedObservationAngle("fwhm", jitter_sigma, obstruction)
psf_jitter = self.__psf_jitter
x = np.arange(psf_jitter.shape[0]) * reduced_observation_angle_pixel.value / self.__osf
psf_interp = interp1d(x=x, y=psf_jitter, kind='cubic', copy=False, bounds_error=False,
fill_value="extrapolate")
res = psf_interp(dist)
res = self._rebin(res, 1 / self.__osf)
res = (mask_red * res).view(np.ndarray)
# Integrate the reduced mask and divide by the indefinite integral to get relative intensities
res = res * (reduced_observation_angle_pixel.value / self.__osf) ** 2 / (4 / np.pi) * (1 - obstruction)
# reintegrate the reduced mask into the complete mask
mask[y_ind.min():(y_ind.max() + 1), x_ind.min():(x_ind.max() + 1)] = res
return mask